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The maximum maximum of a martingale with given $n$ marginals


  • Pierre Henry-Labord`ere
  • Jan Ob{l}'oj
  • Peter Spoida
  • Nizar Touzi


We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

Suggested Citation

  • Pierre Henry-Labord`ere & Jan Ob{l}'oj & Peter Spoida & Nizar Touzi, 2012. "The maximum maximum of a martingale with given $n$ marginals," Papers 1203.6877,, revised Jan 2016.
  • Handle: RePEc:arx:papers:1203.6877

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    References listed on IDEAS

    1. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    4. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929,, revised Feb 2013.
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    Cited by:

    1. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Ob{l}'oj, 2016. "Pointwise Arbitrage Pricing Theory in Discrete Time," Papers 1612.07618,, revised Feb 2018.
    2. Johannes Muhle-Karbe & Marcel Nutz, 2016. "A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing," Papers 1612.09152,, revised Jan 2018.
    3. repec:eee:spapps:v:127:y:2017:i:9:p:3005-3013 is not listed on IDEAS
    4. Julio Backhoff Veraguas & Mathias Beiglbock & Martin Huesmann & Sigrid Kallblad, 2017. "Martingale Benamou--Brenier: a probabilistic perspective," Papers 1708.04869,
    5. Mathias Beiglbock & Marcel Nutz & Nizar Touzi, 2015. "Complete Duality for Martingale Optimal Transport on the Line," Papers 1507.00671,, revised Jun 2016.
    6. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545,
    7. Mathias Beiglboeck & Pierre Henry-Labordere & Nizar Touzi, 2017. "Monotone Martingale Transport Plans and Skorohod Embedding," Papers 1701.06779,
    8. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    9. Marcel Nutz & Florian Stebegg & Xiaowei Tan, 2017. "Multiperiod Martingale Transport," Papers 1703.10588,
    10. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227,, revised Feb 2014.
    11. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    12. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505,
    13. Marcel Nutz & Florian Stebegg, 2016. "Canonical Supermartingale Couplings," Papers 1609.02867,, revised Nov 2017.
    14. Stefan Gerhold & I. Cetin Gulum, 2016. "Consistency of option prices under bid-ask spreads," Papers 1608.05585,

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